Puzzle to illustrate the power of compounding

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

26 November 2023

Suppose you are given the choice of either getting ten million shillings at once today or start with one shilling and then the amount is doubled every day for the next 30 days, which would you choose? In the second option, you get one shilling on the first day, two on the second, four on the third, eight on the fourth and so on up to the 30th day.

A puzzle similar to this one has been doing the round on the social media and David Maina forwarded it to me recently asking whether it is true that the second option far better than the first. Well, the proof of the pudding is in the eating.

This is a simple, albeit tedious, sum to do. The best way to go about it is to draw a table with three columns and thirty rows. On the first column you enter the number of days: from 1 all the way to 30. On the second column will be the amounts received: starting with Sh1, then Sh2, followed by Sh4, then Sh8… Be sure to have a calculator at hand.

The third column is for the running totals. Thus, on day 1, the total received is Sh1; on day 2 it is Sh1 + Sh2 = Sh3; then on the third day it will be Sh3 + Sh4 = Sh4 and so on. If you do the math correctly, you should find that on the last day, day 30, the amount received will be Sh536,870,912. Yes: over half a billion shillings!

And this is just the amount you get on that day; not the running total! When all the amounts are added up from the first to the last day, it comes to Sh1,073,741,823 – over one billion billings.

This puzzle is good illustration of the power of compounding as well as the human brain’s inability to comprehend it. When we hear that something is “doubled every day”, we imagine that it is increased by two units daily. Thus, in the above puzzle, we think the progression will go this way: Sh1, Sh2, Sh4, Sh6, Sh8…Sh60. But, of course, that is not doubling; it is adding two.

For the same reason, it doesn’t sound plausible that, when an investment pays 10 per cent per year (compounded), it will double in vale in slightly over seven years. The expectation is that the doubling will only come after ten years.